Exploiting the structure effectively and efficiently in low rank matrix recovery

TL;DR

This survey reviews recent advances in low rank matrix recovery, covering matrix sensing, completion, and phase retrieval, highlighting effective algorithms and their theoretical guarantees.

Contribution

It provides a comprehensive overview of recent methods and theoretical results in low rank matrix recovery across multiple scenarios.

Findings

Nuclear norm minimization is effective for low rank recovery.

Projected gradient descent based on matrix factorization shows promising results.

Riemannian optimization offers efficient solutions with strong guarantees.

Abstract

Low rank model arises from a wide range of applications, including machine learning, signal processing, computer algebra, computer vision, and imaging science. Low rank matrix recovery is about reconstructing a low rank matrix from incomplete measurements. In this survey we review recent developments on low rank matrix recovery, focusing on three typical scenarios: matrix sensing, matrix completion and phase retrieval. An overview of effective and efficient approaches for the problem is given, including nuclear norm minimization, projected gradient descent based on matrix factorization, and Riemannian optimization based on the embedded manifold of low rank matrices. Numerical recipes of different approaches are emphasized while accompanied by the corresponding theoretical recovery guarantees.